Subgroups — C₁₃₁×#33292 #45719176

Order 33536.

662 conjugacy classes of subgroups.

The subgroup lattice has 318 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#2660×C₁₃₁	C₂	1	2	yes	maximal
2	#1993×C₁₃₁	C₂	1	2	yes	maximal
3	#1992×C₁₃₁	C₂	1	2	yes	maximal
4	#1991×C₁₃₁	C₂	1	2	yes	maximal
5	#1990×C₁₃₁	C₂	1	2	yes	maximal
6	#1836×C₁₃₁	C₂	1	2	yes	maximal
7	#1835×C₁₃₁	C₂	1	1	yes	maximal, characteristic
8	#1834×C₁₃₁	C₂	1	1	yes	maximal, characteristic
9	#1726×C₁₃₁	C₂	1	1	yes	maximal, characteristic
10	C₂×C₄⋊Q₈×C₁₃₁	C₂²	1	2	yes	—
11	C₂×#125×C₁₃₁	C₂²	1	2	yes	—
12	C₂×#97×C₁₃₁	C₂²	1	2	yes	—
13	C₂×#97×C₁₃₁	C₂²	1	1	yes	characteristic
14	C₂×C₄×C₈×C₁₃₁	C₂²	1	2	yes	—
15	C₂×C₄×C₈×C₁₃₁	C₂²	1	1	yes	characteristic
16	#398×C₁₃₁	—	2	2	no	—
17	#398×C₁₃₁	—	2	2	no	—
18	#395×C₁₃₁	—	2	2	no	—
19	#395×C₁₃₁	—	2	2	no	—
20	#384×C₁₃₁	C₂²	1	2	yes	—
21	#384×C₁₃₁	C₂²	1	2	yes	—
22	#382×C₁₃₁	C₂²	1	2	yes	—
23	#382×C₁₃₁	C₂²	1	2	yes	—
24	#376×C₁₃₁	C₂²	1	1	yes	characteristic
25	#340×C₁₃₁	C₂²	1	1	yes	characteristic
26	#340×C₁₃₁	C₂²	1	2	yes	—
27	#340×C₁₃₁	C₂²	1	2	yes	—
28	#340×C₁₃₁	C₂²	1	2	yes	—
29	#340×C₁₃₁	C₂²	1	1	yes	characteristic
30	#340×C₁₃₁	C₂²	1	2	yes	—
31	#340×C₁₃₁	C₂²	1	2	yes	—
32	#340×C₁₃₁	C₂²	1	1	yes	characteristic
33	#340×C₁₃₁	C₂²	1	2	yes	—
34	#340×C₁₃₁	C₂²	1	1	yes	characteristic
35	C₂²×Q₈×C₁₃₁	—	2	2	no	—
36	C₂²×C₈×C₁₃₁	C₂³	1	1	yes	characteristic
37	C₂²×C₈×C₁₃₁	C₂³	1	2	yes	—
38	C₂²×C₈×C₁₃₁	C₂³	1	1	yes	characteristic
39	C₄⋊Q₈×C₁₃₁	—	2	4	no	—
40	C₄⋊Q₈×C₁₃₁	—	2	4	no	—
41	#125×C₁₃₁	—	2	4	no	—
42	#125×C₁₃₁	—	2	4	no	—
43	C₂×C₄⋊C₄×C₁₃₁	—	2	2	no	—
44	C₂×C₄⋊C₄×C₁₃₁	—	2	2	no	—
45	C₂×C₄⋊C₄×C₁₃₁	—	2	2	no	—
46	C₂×C₄⋊C₄×C₁₃₁	—	2	2	no	—
47	C₂×C₄⋊C₄×C₁₃₁	—	2	2	no	—
48	C₂×C₄⋊C₄×C₁₃₁	—	2	2	no	—
49	C₂×C₄⋊C₄×C₁₃₁	—	2	2	no	—
50	C₂×C₄⋊C₄×C₁₃₁	C₂³	1	2	yes	—
51	C₂×C₄⋊C₄×C₁₃₁	C₂³	1	2	yes	—
52	C₂×C₄⋊C₄×C₁₃₁	C₂³	1	2	yes	—
53	C₂×C₄⋊C₄×C₁₃₁	C₂³	1	2	yes	—
54	C₂×C₄²×C₁₃₁	C₂³	1	1	yes	characteristic
55	C₂×C₄²×C₁₃₁	C₂³	1	2	yes	—
56	#97×C₁₃₁	—	2	4	no	—
57	#97×C₁₃₁	Q₈	1	4	yes	complemented
58	C₄×C₈×C₁₃₁	—	2	4	no	—
59	C₄×C₈×C₁₃₁	Q₈	1	4	yes	complemented
60	#95×C₁₃₁	—	2	2	no	—
61	#95×C₁₃₁	—	2	2	no	—
62	#95×C₁₃₁	—	2	2	no	—
63	#95×C₁₃₁	—	2	2	no	—
64	#95×C₁₃₁	D₄	1	1	yes	characteristic
65	#95×C₁₃₁	D₄	1	2	yes	—
66	#95×C₁₃₁	D₄	1	1	yes	characteristic
67	C₂×Q₈×C₁₃₁	—	4	4	no	—
68	C₂×Q₈×C₁₃₁	—	8	2	no	—
69	C₂×Q₈×C₁₃₁	—	8	2	no	—
70	C₂²×C₄×C₁₃₁	—	2	2	no	—
71	C₂²×C₄×C₁₃₁	—	2	2	no	—
72	C₂²×C₄×C₁₃₁	—	2	2	no	—
73	C₂²×C₄×C₁₃₁	—	2	2	no	—
74	C₂²×C₄×C₁₃₁	C₂×D₄	1	2	yes	—
75	C₂²×C₄×C₁₃₁	C₂×D₄	1	2	yes	—
76	C₂²×C₄×C₁₃₁	C₂×D₄	1	1	yes	characteristic
77	C₂²×C₄×C₁₃₁	C₂×D₄	1	1	yes	characteristic
78	C₂²×C₄×C₁₃₁	C₂⁴	1	1	yes	characteristic
79	C₂×C₈×C₁₃₁	—	2	2	no	—
80	C₂×C₈×C₁₃₁	—	2	2	no	—
81	C₂×C₈×C₁₃₁	—	2	2	no	—
82	C₂×C₈×C₁₃₁	—	2	2	no	—
83	C₂×C₈×C₁₃₁	C₂×Q₈	1	2	yes	—
84	C₂×C₈×C₁₃₁	C₂×Q₈	1	2	yes	—
85	C₂×C₈×C₁₃₁	C₂×Q₈	1	4	yes	—
86	C₄⋊C₄×C₁₃₁	—	2	2	no	—
87	C₄⋊C₄×C₁₃₁	—	2	2	no	—
88	C₄⋊C₄×C₁₃₁	—	2	2	no	—
89	C₄⋊C₄×C₁₃₁	—	2	2	no	—
90	C₄⋊C₄×C₁₃₁	—	2	2	no	—
91	C₄⋊C₄×C₁₃₁	—	2	2	no	—
92	C₄⋊C₄×C₁₃₁	—	2	2	no	—
93	C₄⋊C₄×C₁₃₁	—	2	2	no	—
94	C₄⋊C₄×C₁₃₁	—	4	4	no	—
95	C₄⋊C₄×C₁₃₁	—	4	4	no	—
96	C₄⋊C₄×C₁₃₁	—	4	4	no	—
97	C₄⋊C₄×C₁₃₁	—	8	2	no	—
98	C₄⋊C₄×C₁₃₁	—	8	2	no	—
99	C₄⋊C₄×C₁₃₁	—	8	2	no	—
100	C₄⋊C₄×C₁₃₁	—	8	2	no	—
101	C₄²×C₁₃₁	C₂×Q₈	1	2	yes	—
102	C₄²×C₁₃₁	C₂×Q₈	1	2	yes	—
103	C₄²×C₁₃₁	#41	1	4	yes	—
104	C₄²×C₁₃₁	#41	1	4	yes	—
105	C₂³×C₁₃₁	C₂²×D₄	1	1	yes	characteristic
106	Q₈×C₁₃₁	—	8	8	no	—
107	C₂×C₄×C₁₃₁	—	2	1	no	—
108	C₂×C₄×C₁₃₁	—	2	2	no	—
109	C₂×C₄×C₁₃₁	—	2	2	no	—
110	C₂×C₄×C₁₃₁	—	2	2	no	—
111	C₂×C₄×C₁₃₁	—	2	2	no	—
112	C₂×C₄×C₁₃₁	—	2	2	no	—
113	C₂×C₄×C₁₃₁	—	2	1	no	—
114	C₂×C₄×C₁₃₁	—	4	2	no	—
115	C₂×C₄×C₁₃₁	—	4	2	no	—
116	C₂×C₄×C₁₃₁	—	4	2	no	—
117	C₂×C₄×C₁₃₁	—	4	2	no	—
118	C₂×C₄×C₁₃₁	—	4	2	no	—
119	C₂×C₄×C₁₃₁	—	4	2	no	—
120	C₂×C₄×C₁₃₁	—	4	2	no	—
121	C₂×C₄×C₁₃₁	—	4	2	no	—
122	C₂×C₄×C₁₃₁	—	4	2	no	—
123	C₂×C₄×C₁₃₁	—	4	2	no	—
124	C₂×C₄×C₁₃₁	—	4	2	no	—
125	C₂×C₄×C₁₃₁	—	4	2	no	—
126	C₂×C₄×C₁₃₁	#124	1	4	yes	—
127	C₂×C₄×C₁₃₁	#124	1	4	yes	—
128	C₂×C₄×C₁₃₁	C₄⋊Q₈	1	2	yes	—
129	C₂×C₄×C₁₃₁	C₄⋊Q₈	1	2	yes	—
130	C₂×C₄×C₁₃₁	C₂²×Q₈	1	1	yes	characteristic
131	C₂×C₄×C₁₃₁	C₂²×Q₈	1	1	yes	characteristic
132	C₂×C₄×C₁₃₁	C₂×#41	1	2	yes	—
133	C₂×C₄×C₁₃₁	C₂×#41	1	2	yes	—
134	C₈×C₁₃₁	—	2	2	no	—
135	C₈×C₁₃₁	—	2	4	no	—
136	C₈×C₁₃₁	—	2	2	no	—
137	C₂²×C₁₃₁	C₂×#124	1	2	yes	—
138	C₂²×C₁₃₁	C₂×C₄⋊Q₈	1	1	yes	characteristic
139	C₂²×C₁₃₁	#576	1	1	yes	characteristic
140	C₂²×C₁₃₁	#577	1	2	yes	—
141	C₂²×C₁₃₁	#578	1	1	yes	characteristic
142	C₄×C₁₃₁	—	2	4	no	—
143	C₄×C₁₃₁	—	2	4	no	—
144	C₄×C₁₃₁	—	2	2	no	—
145	C₄×C₁₃₁	—	2	2	no	—
146	C₄×C₁₃₁	—	8	2	no	—
147	C₄×C₁₃₁	—	8	2	no	—
148	C₄×C₁₃₁	—	8	2	no	—
149	C₄×C₁₃₁	—	8	2	no	—
150	C₄×C₁₃₁	#533	1	1	yes	characteristic
151	C₄×C₁₃₁	#533	1	2	yes	—
152	C₄×C₁₃₁	#533	1	1	yes	characteristic
153	C₂×C₁₃₁	#2862	1	1	yes	characteristic
154	C₂×C₁₃₁	#3136	1	2	yes	—
155	C₂×C₁₃₁	#3137	1	2	yes	—
156	C₂×C₁₃₁	#3163	1	1	yes	characteristic
157	C₂×C₁₃₁	#3164	1	1	yes	characteristic
158	#33292	C₁₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
159	C₁₃₁	#33292	1	1	yes	characteristic, direct product factor, Sylow
160	#2660	C₂×C₁₃₁	1	2	yes	—
161	#1993	C₂×C₁₃₁	1	2	yes	—
162	#1992	C₂×C₁₃₁	1	2	yes	—
163	#1991	C₂×C₁₃₁	1	2	yes	—
164	#1990	C₂×C₁₃₁	1	2	yes	—
165	#1836	C₂×C₁₃₁	1	2	yes	—
166	#1835	C₂×C₁₃₁	1	1	yes	characteristic
167	#1834	C₂×C₁₃₁	1	1	yes	characteristic
168	#1726	C₂×C₁₃₁	1	1	yes	characteristic
169	C₂×C₄⋊Q₈	C₂²×C₁₃₁	1	2	yes	—
170	C₂×#125	C₂²×C₁₃₁	1	2	yes	—
171	C₂×#97	C₂²×C₁₃₁	1	2	yes	—
172	C₂×#97	C₂²×C₁₃₁	1	1	yes	characteristic
173	C₂×C₄×C₈	C₂²×C₁₃₁	1	2	yes	—
174	C₂×C₄×C₈	C₂²×C₁₃₁	1	1	yes	characteristic
175	#398	—	2	2	no	—
176	#398	—	2	2	no	—
177	#395	—	2	2	no	—
178	#395	—	2	2	no	—
179	#384	C₂²×C₁₃₁	1	2	yes	—
180	#384	C₂²×C₁₃₁	1	2	yes	—
181	#382	C₂²×C₁₃₁	1	2	yes	—
182	#382	C₂²×C₁₃₁	1	2	yes	—
183	#376	C₂²×C₁₃₁	1	1	yes	characteristic
184	#340	C₂²×C₁₃₁	1	2	yes	—
185	#340	C₂²×C₁₃₁	1	1	yes	characteristic
186	#340	C₂²×C₁₃₁	1	1	yes	characteristic
187	#340	C₂²×C₁₃₁	1	1	yes	characteristic
188	#340	C₂²×C₁₃₁	1	1	yes	characteristic
189	#340	C₂²×C₁₃₁	1	2	yes	—
190	#340	C₂²×C₁₃₁	1	2	yes	—
191	#340	C₂²×C₁₃₁	1	2	yes	—
192	#340	C₂²×C₁₃₁	1	2	yes	—
193	#340	C₂²×C₁₃₁	1	2	yes	—
194	C₂²×Q₈	—	2	2	no	—
195	C₂²×C₈	C₂³×C₁₃₁	1	1	yes	characteristic
196	C₂²×C₈	C₂³×C₁₃₁	1	2	yes	—
197	C₂²×C₈	C₂³×C₁₃₁	1	1	yes	characteristic
198	C₄⋊Q₈	—	2	4	no	—
199	C₄⋊Q₈	—	2	4	no	—
200	#125	—	2	4	no	—
201	#125	—	2	4	no	—
202	C₂×C₄⋊C₄	—	2	2	no	—
203	C₂×C₄⋊C₄	—	2	2	no	—
204	C₂×C₄⋊C₄	—	2	2	no	—
205	C₂×C₄⋊C₄	—	2	2	no	—
206	C₂×C₄⋊C₄	—	2	2	no	—
207	C₂×C₄⋊C₄	—	2	2	no	—
208	C₂×C₄⋊C₄	—	2	2	no	—
209	C₂×C₄⋊C₄	C₂³×C₁₃₁	1	2	yes	—
210	C₂×C₄⋊C₄	C₂³×C₁₃₁	1	2	yes	—
211	C₂×C₄⋊C₄	C₂³×C₁₃₁	1	2	yes	—
212	C₂×C₄⋊C₄	C₂³×C₁₃₁	1	2	yes	—
213	C₂×C₄²	C₂³×C₁₃₁	1	1	yes	characteristic
214	C₂×C₄²	C₂³×C₁₃₁	1	2	yes	—
215	#97	—	2	4	no	—
216	#97	Q₈×C₁₃₁	1	4	yes	complemented
217	C₄×C₈	—	2	4	no	—
218	C₄×C₈	Q₈×C₁₃₁	1	4	yes	complemented
219	#95	—	2	2	no	—
220	#95	—	2	2	no	—
221	#95	—	2	2	no	—
222	#95	—	2	2	no	—
223	#95	D₄×C₁₃₁	1	1	yes	characteristic
224	#95	D₄×C₁₃₁	1	2	yes	—
225	#95	D₄×C₁₃₁	1	1	yes	characteristic
226	C₂×Q₈	—	4	4	no	—
227	C₂×Q₈	—	8	2	no	—
228	C₂×Q₈	—	8	2	no	—
229	C₂²×C₄	—	2	2	no	—
230	C₂²×C₄	—	2	2	no	—
231	C₂²×C₄	—	2	2	no	—
232	C₂²×C₄	—	2	2	no	—
233	C₂²×C₄	C₂×D₄×C₁₃₁	1	2	yes	—
234	C₂²×C₄	C₂×D₄×C₁₃₁	1	2	yes	—
235	C₂²×C₄	C₂×D₄×C₁₃₁	1	1	yes	characteristic
236	C₂²×C₄	C₂×D₄×C₁₃₁	1	1	yes	characteristic
237	C₂²×C₄	C₂⁴×C₁₃₁	1	1	yes	characteristic
238	C₂×C₈	—	2	2	no	—
239	C₂×C₈	—	2	2	no	—
240	C₂×C₈	—	2	2	no	—
241	C₂×C₈	—	2	2	no	—
242	C₂×C₈	C₂×Q₈×C₁₃₁	1	2	yes	—
243	C₂×C₈	C₂×Q₈×C₁₃₁	1	2	yes	—
244	C₂×C₈	C₂×Q₈×C₁₃₁	1	4	yes	—
245	C₄⋊C₄	—	2	2	no	—
246	C₄⋊C₄	—	2	2	no	—
247	C₄⋊C₄	—	2	2	no	—
248	C₄⋊C₄	—	2	2	no	—
249	C₄⋊C₄	—	2	2	no	—
250	C₄⋊C₄	—	2	2	no	—
251	C₄⋊C₄	—	2	2	no	—
252	C₄⋊C₄	—	2	2	no	—
253	C₄⋊C₄	—	4	4	no	—
254	C₄⋊C₄	—	4	4	no	—
255	C₄⋊C₄	—	4	4	no	—
256	C₄⋊C₄	—	8	2	no	—
257	C₄⋊C₄	—	8	2	no	—
258	C₄⋊C₄	—	8	2	no	—
259	C₄⋊C₄	—	8	2	no	—
260	C₄²	C₂×Q₈×C₁₃₁	1	2	yes	—
261	C₄²	C₂×Q₈×C₁₃₁	1	2	yes	—
262	C₄²	#41×C₁₃₁	1	4	yes	—
263	C₄²	#41×C₁₃₁	1	4	yes	—
264	C₂³	C₂²×D₄×C₁₃₁	1	1	yes	characteristic
265	Q₈	—	8	8	no	—
266	C₂×C₄	—	2	1	no	—
267	C₂×C₄	—	2	2	no	—
268	C₂×C₄	—	2	2	no	—
269	C₂×C₄	—	2	2	no	—
270	C₂×C₄	—	2	2	no	—
271	C₂×C₄	—	2	2	no	—
272	C₂×C₄	—	2	1	no	—
273	C₂×C₄	—	4	2	no	—
274	C₂×C₄	—	4	2	no	—
275	C₂×C₄	—	4	2	no	—
276	C₂×C₄	—	4	2	no	—
277	C₂×C₄	—	4	2	no	—
278	C₂×C₄	—	4	2	no	—
279	C₂×C₄	—	4	2	no	—
280	C₂×C₄	—	4	2	no	—
281	C₂×C₄	—	4	2	no	—
282	C₂×C₄	—	4	2	no	—
283	C₂×C₄	—	4	2	no	—
284	C₂×C₄	—	4	2	no	—
285	C₂×C₄	#124×C₁₃₁	1	4	yes	—
286	C₂×C₄	#124×C₁₃₁	1	4	yes	—
287	C₂×C₄	C₄⋊Q₈×C₁₃₁	1	2	yes	—
288	C₂×C₄	C₄⋊Q₈×C₁₃₁	1	2	yes	—
289	C₂×C₄	C₂²×Q₈×C₁₃₁	1	1	yes	characteristic
290	C₂×C₄	C₂²×Q₈×C₁₃₁	1	1	yes	characteristic
291	C₂×C₄	C₂×#41×C₁₃₁	1	2	yes	—
292	C₂×C₄	C₂×#41×C₁₃₁	1	2	yes	—
293	C₈	—	2	2	no	—
294	C₈	—	2	4	no	—
295	C₈	—	2	2	no	—
296	C₂²	C₂×#124×C₁₃₁	1	2	yes	—
297	C₂²	C₂×C₄⋊Q₈×C₁₃₁	1	1	yes	characteristic
298	C₂²	#576×C₁₃₁	1	1	yes	characteristic
299	C₂²	#577×C₁₃₁	1	2	yes	—
300	C₂²	#578×C₁₃₁	1	1	yes	characteristic
301	C₄	—	2	4	no	—
302	C₄	—	2	4	no	—
303	C₄	—	2	2	no	—
304	C₄	—	2	2	no	—
305	C₄	—	8	2	no	—
306	C₄	—	8	2	no	—
307	C₄	—	8	2	no	—
308	C₄	—	8	2	no	—
309	C₄	#533×C₁₃₁	1	1	yes	characteristic
310	C₄	#533×C₁₃₁	1	2	yes	—
311	C₄	#533×C₁₃₁	1	1	yes	characteristic
312	C₂	#2862×C₁₃₁	1	1	yes	characteristic
313	C₂	#3136×C₁₃₁	1	2	yes	—
314	C₂	#3137×C₁₃₁	1	2	yes	—
315	C₂	#3163×C₁₃₁	1	1	yes	characteristic
316	C₂	#3164×C₁₃₁	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₁₃₁	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₃₁
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₃₁ ◁ #376×C₁₃₁ ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
57. #97×C₁₃₁	Q₈	semidirect
59. C₄×C₈×C₁₃₁	Q₈	semidirect
158. #33292	C₁₃₁	direct (N × H)
159. C₁₃₁	#33292	direct (N × H)
216. #97	Q₈×C₁₃₁	semidirect
218. C₄×C₈	Q₈×C₁₃₁	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#2660×C₁₃₁	2
#1993×C₁₃₁	2
#1992×C₁₃₁	2
#1991×C₁₃₁	2
#1990×C₁₃₁	2
#1836×C₁₃₁	2
#1835×C₁₃₁	2
#1834×C₁₃₁	2
#1726×C₁₃₁	2
#33292	131

Subgroups — C131×#33292 #45719176

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₁×#33292 #45719176